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Mirrors > Home > MPE Home > Th. List > oprabbii | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
oprabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
oprabbii | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ 𝑤 = 𝑤 | |
2 | oprabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓)) |
4 | 3 | oprabbidv 7220 | . 2 ⊢ (𝑤 = 𝑤 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 {coprab 7157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-oprab 7160 |
This theorem is referenced by: oprab4 7240 mpov 7264 dfxp3 7759 tposmpo 7929 addsrpr 10497 mulsrpr 10498 addcnsr 10557 mulcnsr 10558 joinfval2 17612 meetfval2 17626 dfxrn2 35643 |
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